Splitting polytopes

A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This generalizes a...

Authors: Herrmann, Sven
Joswig, Michael
Division/Institute:FB 10: Mathematik und Informatik
Document types:Article
Media types:Text
Publication date:2008
Date of publication on miami:30.11.2008
Modification date:15.04.2015
Source:Münster Journal of Mathematics, 1 (2008), S. 109-142
Edition statement:[Electronic ed.]
DDC Subject:510: Mathematik
License:InC 1.0
Language:English
Format:PDF document
URN:urn:nbn:de:hbz:6-43529463487
Permalink:https://nbn-resolving.de/urn:nbn:de:hbz:6-43529463487
Digital documents:mjm_vol_1_05.pdf

A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope P, the split complex of P. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].